canonicalPhiScaleSequence_first_closure_law
plain-language theorem explainer
The canonical geometric φ-scale sequence admits a first nontrivial closure index: some natural n is the first place the seed scales close. Anyone citing the T6 self-similarity step or the minimal φ hierarchy would use this existence fact. The proof is a one-line reverse application of the equivalence between first-closure laws and the sequence's closedness predicate.
Claim. There exists a natural number $n$ such that the canonical $\varphi$-geometric scale sequence satisfies the first-closure law at $n$: $n$ is the first nontrivial closure index, and the seed scales of that sequence close at $n$.
background
The Unified Forcing Chain module derives T-1 through T8 as inevitabilities from the Recognition Composition Law plus normalization and calibration. T6 is the step that forces $\varphi$ as the self-similar fixed point of the discrete ledger; geometric scale sequences package the $\varphi$-ladder data used there.
A first-closure law on a geometric scale sequence $S$ is a pair of facts at an index $n$: $n$ is the first nontrivial closure index, and the seed scales of $S$ close at $n$. An upstream equivalence identifies existence of such an $n$ with the existing closedness predicate on $S$: $(\exists n,,\text{first-closure at }n)\leftrightarrow S\text{ is closed}$.
The canonical $\varphi$ scale sequence is the distinguished geometric sequence built from the forced $\varphi$ ratio. Its closedness is already recorded upstream; the present statement only rephrases that closedness in first-closure-law language.
proof idea
One-line term proof. Apply the reverse direction of the upstream equivalence that equates existence of a first-closure law on a geometric scale sequence with that sequence being closed, specialized to the canonical $\varphi$ scale sequence. Discharge the closedness hypothesis by the prior lemma that the canonical $\varphi$ scale sequence is closed. No new arithmetic or forcing work occurs here.
why it matters
This pins the canonical minimal hierarchy as "the $\varphi$ geometric sequence with first closure," the packaging used when T6 forces $\varphi$ from self-similarity in the discrete ledger. It sits inside the Complete Inevitability Chain of the Unified Forcing Chain module, where every level from the absolute floor through T8 is derived from the cost foundation (RCL, $F(1)=0$, $F''(1)=1$).
Downstream use is not yet wired in this graph (no recorded dependents). The declaration still matters as the named existence certificate that the canonical ladder closes at a first nontrivial index, so later hierarchy or octave constructions can quote a single Prop rather than re-open closedness. It does not itself force $\varphi$, the eight-tick period, or $D=3$; those remain T6–T8.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.